It's hard to say without looking or having the exact specs, but (in principal) you should be able to install your graphics card in the new machine, assuming you bought the old one in the past ten years.
First thing to do -
Open up the old PC. Take the screws out of the cover on the left side (as you look at the box from the front) and look inside it. The graphics card will be sticking up at right angles from the motherboard, about halfway up at the back - It is the same physical component that you would have connected your monitor too, so find the connector you used to attach the monitor too and look inside the box behind that panel.
To remove the GPU you'll need to unscrew the screw that attaches is to the case (follow the back of the panel you connected the monitor to, the will be a screw at one end). Disconnect any cables that run from the power supply to the card. There will probably be a catch on the PCI-E connecter of your motherboard. Feel along the base of the card where it connects to the motherboard. You'll feel a long plastic connector where the two meet, and at the end furthest from the back (ie towards the front of the case) there may be a clip or similar - On my motherboard it's just a free bit of plastic that clips over the end and you disconnect it by pushing down towards the bottom of the case but there are a few different designs. It shouldn't be hard to disengage, so don't push hard, the part should move with just a little pressure. Once the clip is undone, you can pull the card directly away from the motherboard (hold the card between your fingers and pull 'backwards' along the axis is was sticking up from) and that's it. The card is removed.
Next, open up the new box. You're taking off the same panel and looking for the same connector in the same place. The PCI covers may be blanks that are screwed in (remove them by undoing the screw) or my be physically connected and require you to get a screwdriver and pry them out. Again, don't force them. You need to wiggle it back and forth gently until the blank breaks out. Push the graphics card into the PCI-E slot (you want the longest one). Screw in the screw. Connect any needed power cables - If you didn't disconnect any then don't worry about this. If you aren't sure they connect at the top, front corner (if the monitor connector is the back, the front is towards the front of the case). They can be on either the top or front, but around there. The connector would be a two rows of three or four circular pins, and there may be two connectors. If you do have them then find the cables from your power supply that fit them and plug them in.
Then close up and plug the monitor into the new output. Turn the machine on. If all has gone well then the machine will boot up as normal, picture on the screen etc. Once it's installed, go online and find the latest drivers from the Nvidia website and install them. And then... Success!
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